Volumes of families of semialgebraic sets

Question

Let $f_0(T),f_1(T) \in \mathbb R [T]$ be polynomials. Let $F(T,X):=X^2+f_1(T)X+f_0(T)$. Then the set $M(t):=\{x \in \mathbb R \mid F(t,x) \le 0\}$ is bounded. Its volume $V(t)$ is $\sqrt{\max\{0,f_1(T)^2-4f_0(T)\}}$. In particular this is piecewise continously differentiable and monotonous, and the number of pieces can be bounded from above in terms of the total degree of $F$.

This motivates the following much more general questions:

For $i=1, \dots, r$, let $F_i \in \mathbb{R}[T,X_1, \dots, X_n]$ be polynomials. We assume that for all real $t > 0$, the sets $M(t):=\{x=(x_1, \dots, x_n) \in \mathbb R^n \mid \text{$F_i(t,x_1, \dots, x_n) \le 0$ for all $i=1, \dots, r$}\}$ are bounded.

I am interested in the function $V(t)$ defined as the volume of $M(t)$, for $t > 0$. In particular:

• Is $V(t)$ piecewise continuously differentiable, with finitely many pieces?
• Is $V(t)$ piecewise monotonous, with finitely many pieces?
• If so: Is there a bound on the number of pieces that depends only on $n$, $r$, the total degree of $F_1, \dots, F_r$?

I would also be interested in conditions on the $F_i$ under which there are such uniform bounds on the number of pieces.

Answer 1

A theorem by Jean-Philippe Rolin and Jean-Marie Lion asserts a similar property in the analytic category. See Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Annales de l'institut Fourier (1998) 48 (3), p.755-767.

I quote their abstract : "Let $f(x,y)$ be a positive subanalytic function defined on $\mathbf R^n×\mathbf R^m$. We prove that the integral $\int_{\mathbf R^m}f(x,y)\,dy$ is a log-analytic function of $x$. Let $Y_x$ be a subanalytic family of global subanalytic subsets of the euclidean space $\mathbf R^m$. We deduce from the previous result that the $k$-dimensional volume of $Y_x$ is a log-analytic function of $x$. A corollary is the log-analytic behaviour of the $k$-dimensional density of a $k$-dimensional subanalytic set at any point of its topological closure."